Bulk Universality for Wigner Matrices - Mathematical Physics

Abstract: We consider $N\times N$ Hermitian Wigner random matrices $H$ where theprobability density for each matrix element is given by the density $ ux=e^{- Ux}$. We prove that the eigenvalue statistics in the bulk is given byDyson sine kernel provided that $U \in C^6\RR$ with at most polynomiallygrowing derivatives and $ ux \le C e^{- C |x|}$ for $x$ large. The proof isbased upon an approximate time reversal of the Dyson Brownian motion combinedwith the convergence of the eigenvalue density to the Wigner semicircle law onshort scales.